Did he ask you or tell you? I'm not a PT but work with clients with a regular schedule and occasionally I'll ask if they'd be able to change day/time so I can fit someone else in that can only have that slot. If they can't, no big deal, we continue with the same time as before.
Did he say it was for new clients? I agree he might want a lie-in or an existing client might have changed job and want to change slot.
Just say no. And if he can't adapt, find someone new.
Sounds very odd. I'm a part time PT and would much rather committed clients than an unknown new client. But yes. I would get rid of him. It's important to have a good relationship with your PT and that is behaviour that suggests he doesn't have your best interests at heart.
He's out of order to move a client with an established schedule and try and bump you for new business. Tell him you need to stick to your 3× per week schedule at 6am. If he is disrespectful keep working out with him while you look for a new PT - is he independent or does he have a boss you can complain to?